Problem 14 - Thomas Calculus

Question

In Exercises \(13-18\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \begin{array}{ll}{\text { a. } 0 \leq x \leq 1} & {\text { b. } 0 \leq x \leq 1, \quad 0 \leq y \leq 1} \\ {\text { c. } 0 \leq x \leq 1,} & {0 \leq y \leq 1, \quad 0 \leq z \leq 1}\end{array} $$

Step-by-Step Solution

Answer

a) Line segment; b) Square; c) Cube.

Step 1: Understand the problem

We need to describe the sets of points in three-dimensional space that satisfy given inequalities for coordinates x, y, and z.

Step 2: Describe set for a.

The inequality for part (a) is given as \(0 \leq x \leq 1\). This represents a line segment along the x-axis from point (0,0,0) to point (1,0,0) in a 3D space.

Step 3: Describe set for b.

For part (b), the inequalities are \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). Together these describe a square in the xy-plane spanning from (0,0,0) to (1,1,0). This square has vertices at (0,0), (1,0), (0,1), and (1,1) with constant z=0.

Step 4: Describe set for c.

In part (c), we have three inequalities: \(0 \leq x \leq 1\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 1\). These describe a cube in 3D space, or more specifically, the unit cube, bounded by the planes x=0, x=1, y=0, y=1, z=0, and z=1.

Key Concepts

Inequalities

An inequality in mathematics expresses a relationship between quantities where they might not be exactly equal, but instead show how one quantity compares to another. Inequalities use symbols such as:

• The "less than" symbol (<)
• The "greater than" symbol (>)
• Or "less than or equal to" (\(\leq\)) and "greater than or equal to" (\(\geq\)).

When dealing with coordinates in a geometric space, inequalities define a range of values that these coordinates can take.
For example, the inequality \(0 \leq x \leq 1\) limits the x-coordinate to be in the range from 0 to 1.
Thus, inequalities help us describe a range or region within a geometric space, such as lines, planes, or even volumes, based on these limitations. In 3D geometry, inequalities are crucial in defining boundaries for geometric shapes like cubes or rectangular prisms.

Coordinate Systems

A coordinate system is a framework used in geometry to uniquely determine the position of points or other geometric elements in space. The most common is the Cartesian coordinate system, consisting of three axes in 3D geometry: x, y, and z.
These axes intersect at a point called the origin (0,0,0). Using these axes, we can provide an ordered set of values, often called coordinates, to specify the location of any point in space.

• The x-axis measures horizontal distance.
• The y-axis measures vertical distance.
• The z-axis measures depth.

By providing inequalities like \(0 \leq x \leq 1\), we define specific regions in this coordinate system.
For example, when only x is limited, we have a line, while limiting both x and y outlines a plane or a surface. This systematic approach using coordinates allows us to map out 3D spaces with precision, making it an essential tool in mathematics and in fields like engineering, physics, and computer graphics.

Unit Cube

The unit cube is an important concept in 3D geometry. It is a cube whose sides each measure 1 unit in length, meaning its volume is exactly one cubic unit. In the context of coordinate systems, its boundaries are defined by the inequalities \(0 \leq x \leq 1\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 1\).
These inequalities specify that the unit cube extends over:

• From x = 0 to x = 1,
• From y = 0 to y = 1,
• From z = 0 to z = 1.

This creates a cube in the 3D space bounded by six planes (x=0, x=1, y=0, y=1, z=0, and z=1). The unit cube is a foundational element in geometry and calculus, serving as a basic building block for more complex shapes and volumes.
It also plays a vital role in various applications, such as in integration over a volume within a 3D space, and serves as a base case in theoretical proofs.